Polar Coordinates — Question 1

Question 1

Problem

The point has polar coordinates $(-4,\pi/3)$. Give two equivalent representations with positive radius and locate its quadrant.

Question 1 – Solution

Solution

1. Use the polar–Cartesian relationships $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2.$$ Equivalent polar coordinates satisfy $$(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+(2k+1)\pi).$$

2. Apply the identity that matches the requested conversion, symmetry test, or intersection, and then check the resulting point or curve in the original polar equation.

3. Adding $\pi$ while changing the radius sign gives $(4,4\pi/3)$; also $(4,-2\pi/3)$.

4. It lies in Quadrant III.